Spectral properties of partial automorphisms of binary rooted tree

TL;DR

This paper investigates the spectral distribution of random partial automorphisms of binary rooted trees, showing that as the tree size grows, the eigenvalues concentrate at zero.

Contribution

It introduces the asymptotic spectral behavior of partial automorphisms in rooted trees, revealing convergence of eigenvalue distributions to a delta measure at zero.

Findings

Eigenvalue distribution converges to delta at zero as tree size increases

Spectral measures of random partial automorphisms become degenerate in the limit

Provides asymptotic analysis of automorphism spectra in rooted trees

Abstract

We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism $x$ we assign its action matrix $A_x$. It is shown that the uniform distribution on eigenvalues of $A_x$ converges weakly in probability to $\delta_0$ as $n \to \infty$, where $\delta_0$ is the delta measure concentrated at $0$.